4.3.1 Basic Kelly Ⓜodel

From the papers Niko Marttinen (2001)1919 Kindly refer to 1th paper in Reference for industry knowdelege and academic research portion for the paper. and Jeffrey Alan Logan Snyder (2013)2020 Kindly refer to 2nd paper in Reference for industry knowdelege and academic research portion for the paper. under 7.4 References both applying Full-Kelly,Half-Kelly and also Quarter-Kelly models which similar with my previous Kelly-Criterion model ®γσ, Eng Lian Hu 20142121 Kindly refer to 4th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References which had applied it and generates an impressive return. but enhanced. Niko Marttinen (2001) has concludes that the basic Kelly criterion generates the highest returns in long run compare to fractional Kelly models.

To achieve the level of profitable betting, one must develop a correct money management procedure. The aim for a punter is to maximize the winnings and minimize the losses. If the punter is capable of predicting accurate probabilities for each match, the Edward O. Thorp (2006)2222 Kindly refer to 6th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References has proven to work effectively in betting. It was named after an American economist John Kelly (1956)2323 Kindly refer to 26th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References and originally designed for information transmission. The Kelly criterion is described below:

figure 4.3.1.1 : Kelly criterion formula.

\[S = \frac{\rho_{EM} \times BK_{Decimal\ odds} - 1} {BK_{HK\ odds}} \cdots equation\ 4.3.1.1\]

• Where \(S\) is the stake expressed as a fraction of one’s total bankroll.
• \(\rho_{EM}\) is probability of an event to take place and
• while \(BK_{Decimal\ odds}\) is decimal odds (decimal odds the return rates with capital stakes) and \(BK_{HK\ Odds}\) (HK odds is the net profit rates without capital stakes) for an event offered by the bookmaker.

Due to HK odds or decimal odds start from range \((0,\infty]\) and return will be \([0,\infty]\), therefore logarithmic function required. For Malay odds \([-1,1]\) no need logarithm. Here I switch from equation 4.3.1.1 to equation 4.3.1.2 as below.

\[log(S) = log(\rho_{EM}) + log(BK_{Decimal\ odds} - 1) - log(BK_{HK\ odds}) \cdots equation\ 4.3.1.2\]

Three important properties, mentioned by Hausch and Ziemba (1994)2424 You can refer to Efficiency of Racetrack Betting Markets (2008 Preface Edition) which is 29th paper in Reference for industry knowdelege and academic research portion for the paper. or Chapter 18 Efficiency of Sports and Lottery Betting Markets in FINANCE for further study about Hausch and Ziemba’s researchs. arise when using this criterion to determine a proper stake for each bet:

• It maximizes the asymptotic growth rate of capital
• Asymptotically, it minimizes the expected time to reach a specified goal
• It outperforms in the long run any other essentially different strategy almost surely

figure 4.3.1.2 : Example of application Kelly criterion.

The criterion is known to economists and financial theorists by names such as the geometric mean maximizing portfolio strategy, the growth-optimal strategy, the capital growth criterion, etc. We will now show that Kelly betting will maximize the expected log utility for sports-book betting.

The rRates value from table above excludes the Cancelled bets. By refer to equation 4.3.1.2, now we fit the adge value from equation 4.1.1 into it to get the rEMProbB2 and rEMProbL2 with known staked value \(S\)2626 Although the result will not be accurate due to the we mention at first, the firm A will not only place bets via only agent A. Let say edge of 0.10 and 0.20 also placed maximum bet HKD40000 but the firm A might placed different amount through other agency based on different edge. However based on the stakes we can reverse the optimal EM Odds. to replace the existing EM value. 2727 Initially think of linear modelling and get the mean value, the positive standard deviation value will be counted as edge range and the residuals value will be the different within the stakes across the leagues. It will similar with proportional staking model as states in paper Good and bad properties of the Kelly criterion by MacLean, Thorp and Ziemba (2010) and concludes that the Full-Kelly model is the best model for long run, you can refer to the reference in Kelly Criterion - Part II for further understanding.

\[log(\rho_{EM}) = log(S) + log(BK_{HK\ odds} + 1) - log(BK_{Decimal\ odds}) \cdots equation\ 4.3.1.3\]

Although the Kelly model is very simple, but we need to seperates the staking based on different leagues or time range to make it applicable to real world. I don’t pretend to know the correct model again but guess the applicable model by testing few models and choose the best among them.

We try to apply the equation 4.3.1.3 to get the Kelly stakes for every single soccer match.

Due to there have few reference papers conducting few staking strategics and concludes full Kelly model is the best fit and most profitable along the long term investment, here I try to simuulate the half-Kelly and also quadruple-Kelly, as well as double-Kelly staking model etc and get the optimal weighted control parameter.2828 There has a reference paper in section 2 of Application of Kelly Criterion model in Sportsbook Investment has compare few models as well and also provides the pro-and-con of Kelly model in investment. However, Kelly model will be the best across the long term investment. Besides, there have few papers doing research and also critic on the Kelly model in investment in financial market and also betting market (includes the rebates of the credit market as well), PIMCO’s fund manager Bill Gross who manage more than one trillion USD funds applied Kelly model for portfolio, George Soros and Warren Buffet also applied similar theoty or method with Kelly although there has no evidence to proof it. You are feel free to know in later section 4.5 Staking Ⓜodel and Ⓜoney Ⓜanagement. For further details kindly refer to Application of Kelly Criterion model in Sportsbook Investment.

Fractional Kelly models are the weight function for Kelly criterion When we talk about weight function in Kelly model. A Response to Professor Paul A Samuelson’s Objections to Kelly Capital Growth Investing has talk about the investment portfolio and compare the double-Kelly, full-Kelly, half-Kelly, quadruple-Kelly and also proportional betting across different stages of iterations and concludes that the full-Kelly will be the best fit and growth beyond the ages. Well, fractional-Kelly (means double-Kelly, half-Kelly and quadruple-Kelly but not full-Kelly model) models will be elastics and lesser will be more conservative and double-Kelly will be very risky and eventually going to bankcrupt due to the staking leverages ratio is twice of full-Kelly and over the sustainability of capital. and For further details kindly refer to Application of Kelly Criterion model in Sportsbook Investment. Therefore in last basic Kelly we use the full-Kelly within same leagues but due to there has different levels of risk setting across different soccer leagues. Therefore a weight function needed to make the staking strategy flexible, and it is term as Kelly portfolio to diversified the investment.

4.3.2 Fractional Kelly Ⓜodel

Now we try to fit a weight function into basic Kelly model to be fractional Kelly model. I try to use log to test the maximum value of weight parameter. You can just simply use \(w = \frac{1}{4}\) or \(log(w) = \frac{1}{2}\) while \(w\) is a vector. Please be mind that the value greater than 1 will be risky since involve leverage and lesser will be more conservative.

\[log(w_{i}) + log(\rho_{i}) \cdots equation\ 4.3.2.1\]

From Niko Marttinen (2001), we can know the full-Kelly generates couple times profit compare to fractional Kelly-models. However there has two points need to be enhanced.

• The high risk at the beginning period of investment.
• Test on different level of edge and concludes that the 145% generates the highest return.

Below Fabián Enrique Moya (2012) also test the fractional Kelly models with diversify money management methods.

table 4.3.2.1 : 177 x 6 : League stakes profiling of firm A year 2011~2015.

Above league risk profile suppose to stores the maximum bet for every single league but I only randomly select 6 leagues as sample. However due to I’ve not yet write a function for real time API2929 There are a lot of real time XML odds price and staking softwares similar with 4lowin2 which was states at the begining section in Part I with operators and test the maximum stakes per bet therefore here I reverse the both mean and median value as the baseline stakes for every single league with a certain range of standard deviation for monte carlo simulation in later section.

vKelly2(dat)$Kelly1$data %>% select(rEMProbB, netProbB, netEMEdge) %>% colMeans
# rEMProbB  netProbB netEMEdge 
# 0.5165538 0.5056328 1.0216569 
#'@  0.5165538 / 0.5056328
[1] 1.021599

# 2.62% (rank: 18)
# 3.09% (rank: 04)
# 3.09% (rank: 03)
#'@ 2.62 * 1.021599
[1] 2.676589

4.3.3 Dynamic Kelly Ⓜodels

In previous section I measure the data from 2011~2015 as static analysis. Well, now I try to seperates as annum data base and get the optimal weight value for next year use. Due to I dont know if the weight function is needed for staking models since sports consultancy firm had applied Poison models weith weight function. I don’t pretand to know and only simulate it by obseravation to guess the optimal value.

Due to fractional Kelly model is independent models (for example : half-Kelly will be half-Kelly staking model, and full-Kelly will be only full-Kelly model across the years as we made comparison in section [4.3.2 Fractional Kelly odel].), now we need to make it dynamic fractional model. Similar with my prevous Rmodel which applied on Poisson model. Due to the calculation of the settlement and result on the win and loss of Asian Handicap is different with Fixed odds, the probabilities of the outcome will be descrete and the measurement of the likelihood result required in order to maximize the profit. here we need to add an additional parameter controller to adjust the staking amount on every single match.

Now I try to will simulate an enhanced Kelly model on staking which take the effect of the outcome of the result into calculation below controller parameter \(\phi(r)\) fit into \(equation\ 4.3.3.1\) to control the leverage ratio.3030 similar theory apply on investment portfolio while it might turn to be nested controller parameters across different soccer leagues.

\[\phi(r) = exp(w_{i}\rho_{i}) \cdots equation\ 4.3.3.1\] Where \(X = x_{i,2,3...n}\) is the original staking amount by Kelly model. Meanwhile, the \(r\) value is the optimal parameter controller for staking.

\[r\begin{Bmatrix} =& Win\\ =& Half\ Win\\ =& Push\\ =& Half\ Loss\\ =& Loss\end{Bmatrix} \cdots equation\ 4.3.3.2\]

## I dont pretand to know the true weight function, here I try both methods bewlo to do the comparison.
## option 1
- $\theta$ = 1    if Win
           = 0.5  if Win-Half
           = 0    if push
           = -0.5 if Loss-Half
           = -1    if Loss
## option 2
- 5 dummy variables which are :
           win = ifelse(1, 0)
           win-half = ifelse(1, 0)
           push = ifelse(1, 0)
           loss-half = ifelse(1, 0)
           loss = ifelse(1, 0)

## Need to correct the content, below just a sample...

## Random pick 2 observations among each stratified levels of Result.
K <- llply(split(dat, dat$Result), function(x) x[sample(nrow(x), 2), c('Result', 'Return', 'EUPrice', 'HKPrice', 'Stakes', 'Rebates')]) %>% ldply(., .id = 'Result') %>% mutate(KReturn = currency(Return), Return = currency(Return), KStakes = currency(Stakes), Stakes = currency(Stakes), Rebates = percent(Rebates), Change = percent(KStakes / Stakes - 1)) %>% .[sample(nrow(.), nrow(.)), ]

## http://www.w3schools.com/colors/colors_picker.asp
K %>% formattable(list(
    Result = formatter('span', style = x ~ ifelse(x == 'Win', style(color = '#269900', font.weight = 'bold'), ifelse(x == 'Half Win', style(color = '#40FF00'), ifelse(x == 'Push', style(color = '#FFFF00'), ifelse(x == 'Half Loss', style(color = '#FF8C1A'), ifelse(x == 'Loss', style(color = '#FF0000', font.weight = 'bold'), NA)))))),
    KReturn = formatter('span', style = ~ style(color = ifelse(KReturn >= KStakes, 'green', 'red')), ~ icontext(ifelse(KReturn >= KStakes, 'plus-sign', 'minus-sign'), KReturn)),
    Return = formatter('span', style = ~ style(color = ifelse(Return >= Stakes, 'green', 'red')), ~ icontext(ifelse(Return >= Stakes, 'plus-sign', 'minus-sign'), Return)),
    EUPrice = color_tile('white', '#003D99'),
    HKPrice = color_tile('white', '#003D99'),
    KStakes = color_tile('white', '#CC9900'),
    Stakes = color_tile('white', '#CC9900'),
    
    Change = formatter('span', style = ~ style(color = ifelse(Change < 0, 'red', 'green')), ~ icontext(ifelse(Change < 0, 'arrow-down', 'arrow-up'), Change))))

table 4.3.3.1 : 12 x 9 : Kelly stakes and firm A’s stakes of annually investment summary table.3131 The annum return rates but not all data return rates since we need to use the return of that particular year to reverse. The weight parameter in later section will take last year performance to be weight controller. Due to the firm A does not only placed bets with one agent but couples of agencies. Let say placed one millions HKD on a soccer match with higher price simultaniously compare to placed bets on one agency with price decreasingly, therefore the Kelly stakes will definately defferent with firm A’s stakes.

- in order to save the executive time, here I load the saved RData file which has simulate to get the optimal weight value. Now I try to fit the weight parameter into the Kelly model to get the result and doing comparison as below.